Einstein manifold

In

If M is the underlying n-dimensional manifold, and g is its metric tensor, the Einstein condition means that

${\displaystyle \mathrm {Ric} =kg}$

for some constant k, where Ric denotes the

Taking the trace of both sides reveals that the constant of proportionality k for Einstein manifolds is related to the scalar curvature R by

${\displaystyle R=nk,}$

where n is the dimension of M.

In

• All 2D manifolds are trivially Einstein manifolds. This is a result of the Riemann tensor having a single degree of freedom.
• Any manifold with
  constant sectional curvature
  is an Einstein manifold—in particular:
  • Euclidean space, which is flat, is a simple example of Ricci-flat, hence Einstein metric.
  • The n-sphere, ${\displaystyle S^{n}}$, with the round metric is Einstein with ${\displaystyle k=n-1}$.
  • Hyperbolic space with the canonical metric is Einstein with ${\displaystyle k=-(n-1)}$.
• Complex projective space, ${\displaystyle \mathbf {CP} ^{n}}$, with the Fubini–Study metric, have ${\displaystyle k=n+1.}$
• Kähler
  , with Einstein constant ${\displaystyle k=0}$. Such metrics are not unique, but rather come in families; there is a Calabi–Yau metric in every Kähler class, and the metric also depends on the choice of complex structure. For example, there is a 60-parameter family of such metrics on K3, 57 parameters of which give rise to Einstein metrics which are not related by isometries or rescalings.
• Kähler–Einstein metrics exist on a variety of compact complex manifolds due to the existence results of Shing-Tung Yau, and the later study of K-stability especially in the case of Fano manifolds.
• An Einstein–Weyl geometry is a generalization of an Einstein manifold for a Weyl connection of a conformal class, rather than the Levi-Civita connection of a metric.

A necessary condition for

• ISBN 3-540-74120-8
  .